The superior biocompatibility and biodegradability of polyhydroxyalkanoates (PHAs) compared to man-made biopolymers such as polylactic acid promise huge potential in biomedical applications, especially tissue engineering (TE). Textile fiber-based TE scaffolds offer unique opportunities to imitate the anisotropic, hierarchical, or strain-stiffening properties of native tissues. A combination of PHAs' enhanced biocompatibility and fiber-based TE scaffolds could improve the performance of TE scaffolds. However, the PHAs' complex crystallization behavior and the resulting intricate spinning procedures remain a challenge. This review focuses on discussing the developments in PHA melt and wet spinning, their challenges, process parameters, and fiber characteristics while revealing the lack of an in-depth fiber characterization of wet-spun fibers compared to melt-spun filaments, leading to squandered potential in scaffold development. Additionally, the biomedical application of PHAs other than poly-4-hydroxybutyrate is hampered by a failure of polymer purity to meet the requirements for biomedical applications.